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Course: Statistics and probability > Unit 3

Lesson 7: Box and whisker plots
Math
Statistics and probability
Summarizing quantitative data
Box and whisker plots
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Identifying outliers with the 1.5xIQR rule

Google Classroom
An outlier is a data point that lies outside the overall pattern in a distribution.
The distribution below shows the scores on a driver's test for 19 applicants. How many outliers do you see?
A dot plot has a horizontal axis labeled scores numbered from 0 to 25. Dots are plotted above the following: 5, 1; 7, 1; 10, 1; 15, 1; 19, 1; 21, 2; 22, 2; 23, 5; 24, 4; 25, 1.
Some people may say there are 5 outliers, but someone else might disagree and say there are 3 or 4 outliers. Statisticians have developed many ways to identify what should and shouldn't be called an outlier.
A commonly used rule says that a data point is an outlier if it is more than 1.5⋅IQR above the third quartile or below the first quartile. Said differently, low outliers are below Q1−1.5⋅IQR and high outliers are above Q3+1.5⋅IQR.
Let's try it out on the distribution from above.

Step 1) Find the median, quartiles, and interquartile range

Here are the 19 scores listed out.
5, 7, 10, 15, 19, 21, 21, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25
What is the median?
median=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

What is the first quartile?
Q1=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

What is the third quartile?
Q3=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

What is the interquartile range?
IQR=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Step 2) Calculate 1.5⋅IQR below the first quartile and check for low outliers.

problem a
Calculate Q1−1.5⋅IQR
Q1−1.5⋅IQR=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

problem b
How many data points can we say are low outliers?
A dot plot has a horizontal axis labeled scores numbered from 0 to 25. Dots are plotted above the following: 5, 1; 7, 1; 10, 1; 15, 1; 19, 1; 21, 2; 22, 2; 23, 5; 24, 4; 25, 1.
Choose 1 answer:

Step 3) Calculate 1.5⋅IQR above the third quartile and check for high outliers.

problem a
Calculate Q3+1.5⋅IQR
Q3+1.5⋅IQR=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

problem b
How many data points can we say are high outliers?
A dot plot has a horizontal axis labeled scores numbered from 0 to 25. Dots are plotted above the following: 5, 1; 7, 1; 10, 1; 15, 1; 19, 1; 21, 2; 22, 2; 23, 5; 24, 4; 25, 1.
Choose 1 answer:

Bonus learning: Showing outliers in box and whisker plots

Box and whisker plots will often show outliers as dots that are separate from the rest of the plot.
Here's a box and whisker plot of the distribution from above that does not show outliers.
A box and whisker plot above a line labeled scores. The left side of the whisker at 5. The beginning part of the box is at 19. The ending part of the box is at 24. The right side of the whisker is at 25.
Here's a box and whisker plot of the same distribution that does show outliers.
Notice how the outliers are shown as dots, and the whisker had to change. The whisker extends to the farthest point in the data set that wasn't an outlier, which was 15.
Here's the original data set again for comparison.
A dot plot has a horizontal axis labeled scores numbered from 0 to 25. Dots are plotted above the following: 5, 1; 7, 1; 10, 1; 15, 1; 19, 1; 21, 2; 22, 2; 23, 5; 24, 4; 25, 1.

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